{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:HL5OZH22ESH374P32ESIGREG43","short_pith_number":"pith:HL5OZH22","canonical_record":{"source":{"id":"2108.04113","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-08-03T13:42:57Z","cross_cats_sorted":[],"title_canon_sha256":"8b8707afe84cf56f7cc6a1ef3e149562832351ab1248d84a18f03736014289a9","abstract_canon_sha256":"964b99133273b24278057d57bcd53ac99c210b55c552aa53bce062a88c08f513"},"schema_version":"1.0"},"canonical_sha256":"3afaec9f5a248fbff1fbd124834486e6d0c3feb7a84c06d538b57ec2cbb347ce","source":{"kind":"arxiv","id":"2108.04113","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2108.04113","created_at":"2026-07-05T04:44:40Z"},{"alias_kind":"arxiv_version","alias_value":"2108.04113v3","created_at":"2026-07-05T04:44:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.04113","created_at":"2026-07-05T04:44:40Z"},{"alias_kind":"pith_short_12","alias_value":"HL5OZH22ESH3","created_at":"2026-07-05T04:44:40Z"},{"alias_kind":"pith_short_16","alias_value":"HL5OZH22ESH374P3","created_at":"2026-07-05T04:44:40Z"},{"alias_kind":"pith_short_8","alias_value":"HL5OZH22","created_at":"2026-07-05T04:44:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:HL5OZH22ESH374P32ESIGREG43","target":"record","payload":{"canonical_record":{"source":{"id":"2108.04113","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-08-03T13:42:57Z","cross_cats_sorted":[],"title_canon_sha256":"8b8707afe84cf56f7cc6a1ef3e149562832351ab1248d84a18f03736014289a9","abstract_canon_sha256":"964b99133273b24278057d57bcd53ac99c210b55c552aa53bce062a88c08f513"},"schema_version":"1.0"},"canonical_sha256":"3afaec9f5a248fbff1fbd124834486e6d0c3feb7a84c06d538b57ec2cbb347ce","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:44:40.468146Z","signature_b64":"afn7EJb9iTUP+97ituhrwoL5fgyxEbAyBtbsaFYm0T1QXfPSn/oWvbWxWAiHmjomk7UYnia+y5xk3vbyMcX/CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3afaec9f5a248fbff1fbd124834486e6d0c3feb7a84c06d538b57ec2cbb347ce","last_reissued_at":"2026-07-05T04:44:40.467787Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:44:40.467787Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2108.04113","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T04:44:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ii0pELnS5fIdQnmf1gwMLpqsHxYP542+VdcipVZyw26XZFxPJCaGACxg1qwYOa1L7bEzDc+gAS8qqmxYb+IsCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T08:35:38.006758Z"},"content_sha256":"1e443a2ccee927aa07f8908655dee9f74711c8e799d157c4322442b51ccfe6e5","schema_version":"1.0","event_id":"sha256:1e443a2ccee927aa07f8908655dee9f74711c8e799d157c4322442b51ccfe6e5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:HL5OZH22ESH374P32ESIGREG43","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Ledin and Brousseau's summation problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kunle Adegoke","submitted_at":"2021-08-03T13:42:57Z","abstract_excerpt":"We develop a recursive scheme, as well as polynomial forms (polynomials in $n$ of degree $m$), for the evaluation of Ledin and Brousseau's Fibonacci sums of the form $S(m,n,r)=\\sum_{k=1}^nk^mF_{k + r}$, $T(m,n,r)=\\sum_{k=1}^nk^mL_{k + r}$ for non-negative integers $m$ and $n$ and arbitrary integer $r$; $F_j$ and $L_j$ being the $j^{th}$ Fibonacci and Lucas numbers. We also extend the study to a general second order sequence by establishing a recursive procedure to determine $W(m,n,r;a,b,p,q)=\\sum_{k=1}^nk^mw_{k+r}$ where $(w_j(a,b;p,q))$ is the Horadam sequence defined by $w_0 = a,\\,w_1 = b;\\,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.04113","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2108.04113/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T04:44:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X/HMPb2pQ7TYFua9Bd3FZHgL8uVBF9vw1tyZBTJ+J2KGNpsox0CsXFJ3rpG99qQApflxPJHDp0jMoZIBb3+vDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T08:35:38.007656Z"},"content_sha256":"d703ad6554c41c7058e1e33fd65907d3912494833af57b6d0747b058f6821063","schema_version":"1.0","event_id":"sha256:d703ad6554c41c7058e1e33fd65907d3912494833af57b6d0747b058f6821063"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HL5OZH22ESH374P32ESIGREG43/bundle.json","state_url":"https://pith.science/pith/HL5OZH22ESH374P32ESIGREG43/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HL5OZH22ESH374P32ESIGREG43/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-12T08:35:38Z","links":{"resolver":"https://pith.science/pith/HL5OZH22ESH374P32ESIGREG43","bundle":"https://pith.science/pith/HL5OZH22ESH374P32ESIGREG43/bundle.json","state":"https://pith.science/pith/HL5OZH22ESH374P32ESIGREG43/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HL5OZH22ESH374P32ESIGREG43/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:HL5OZH22ESH374P32ESIGREG43","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"964b99133273b24278057d57bcd53ac99c210b55c552aa53bce062a88c08f513","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-08-03T13:42:57Z","title_canon_sha256":"8b8707afe84cf56f7cc6a1ef3e149562832351ab1248d84a18f03736014289a9"},"schema_version":"1.0","source":{"id":"2108.04113","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2108.04113","created_at":"2026-07-05T04:44:40Z"},{"alias_kind":"arxiv_version","alias_value":"2108.04113v3","created_at":"2026-07-05T04:44:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.04113","created_at":"2026-07-05T04:44:40Z"},{"alias_kind":"pith_short_12","alias_value":"HL5OZH22ESH3","created_at":"2026-07-05T04:44:40Z"},{"alias_kind":"pith_short_16","alias_value":"HL5OZH22ESH374P3","created_at":"2026-07-05T04:44:40Z"},{"alias_kind":"pith_short_8","alias_value":"HL5OZH22","created_at":"2026-07-05T04:44:40Z"}],"graph_snapshots":[{"event_id":"sha256:d703ad6554c41c7058e1e33fd65907d3912494833af57b6d0747b058f6821063","target":"graph","created_at":"2026-07-05T04:44:40Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2108.04113/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We develop a recursive scheme, as well as polynomial forms (polynomials in $n$ of degree $m$), for the evaluation of Ledin and Brousseau's Fibonacci sums of the form $S(m,n,r)=\\sum_{k=1}^nk^mF_{k + r}$, $T(m,n,r)=\\sum_{k=1}^nk^mL_{k + r}$ for non-negative integers $m$ and $n$ and arbitrary integer $r$; $F_j$ and $L_j$ being the $j^{th}$ Fibonacci and Lucas numbers. We also extend the study to a general second order sequence by establishing a recursive procedure to determine $W(m,n,r;a,b,p,q)=\\sum_{k=1}^nk^mw_{k+r}$ where $(w_j(a,b;p,q))$ is the Horadam sequence defined by $w_0 = a,\\,w_1 = b;\\,","authors_text":"Kunle Adegoke","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-08-03T13:42:57Z","title":"On Ledin and Brousseau's summation problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.04113","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1e443a2ccee927aa07f8908655dee9f74711c8e799d157c4322442b51ccfe6e5","target":"record","created_at":"2026-07-05T04:44:40Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"964b99133273b24278057d57bcd53ac99c210b55c552aa53bce062a88c08f513","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-08-03T13:42:57Z","title_canon_sha256":"8b8707afe84cf56f7cc6a1ef3e149562832351ab1248d84a18f03736014289a9"},"schema_version":"1.0","source":{"id":"2108.04113","kind":"arxiv","version":3}},"canonical_sha256":"3afaec9f5a248fbff1fbd124834486e6d0c3feb7a84c06d538b57ec2cbb347ce","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3afaec9f5a248fbff1fbd124834486e6d0c3feb7a84c06d538b57ec2cbb347ce","first_computed_at":"2026-07-05T04:44:40.467787Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:44:40.467787Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"afn7EJb9iTUP+97ituhrwoL5fgyxEbAyBtbsaFYm0T1QXfPSn/oWvbWxWAiHmjomk7UYnia+y5xk3vbyMcX/CA==","signature_status":"signed_v1","signed_at":"2026-07-05T04:44:40.468146Z","signed_message":"canonical_sha256_bytes"},"source_id":"2108.04113","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1e443a2ccee927aa07f8908655dee9f74711c8e799d157c4322442b51ccfe6e5","sha256:d703ad6554c41c7058e1e33fd65907d3912494833af57b6d0747b058f6821063"],"state_sha256":"330a51c66c0b9db4fcba1ecd701d5311ae8cae21babcb39402109053998c062e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PiP3xN5RKyTjlB5QOqiyt61LJ3A4BZQGeiIG8lpOr244myTL3C6PGTsZynmQi7tkyF/CKfHSSupHW+K7Y+hkBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T08:35:38.013377Z","bundle_sha256":"c0a7eb872bf455807aa4039b5dbf6fe177596839d6250312de6963b63842882c"}}